Semiclassical theory of weak antilocalization and spin relaxation in ballistic quantum dots

Semiclassical theory of weak antilocalization and spin relaxation in ballistic quantum dots

Oleg Zaitsev,^1,*Diego Frustaglia,²and Klaus Richter¹

1Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany

2NEST-INFM and Scuola Normale Superiore, 56126 Pisa, Italy 共Received 7 June 2005; published 25 October 2005兲

We develop a semiclassical theory for spin-dependent quantum transport in ballistic quantum dots. The theory is based on the semiclassical Landauer formula, that we generalize to include spin-orbit and Zeeman interaction. Within this approach, the orbital degrees of freedom are treated semiclassically, while the spin dynamics is computed quantum mechanically. Employing this method, we calculate the quantum correction to the conductance in quantum dots with Rashba and Dresselhaus spin-orbit interaction. We find a strong sensi- tivity of the quantum correction to the underlying classical dynamics of the system. In particular, a suppression of weak antilocalization in integrable systems is observed. These results are attributed to the qualitatively different types of spin relaxation in integrable and chaotic quantum cavities.

DOI:10.1103/PhysRevB.72.155325 PACS number共s兲: 73.23.⫺b, 03.65.Sq, 71.70.Ej

I. INTRODUCTION

Guided by the vision to incorporate spin physics into the far-advanced semiconductor 共hetero兲structure technology, semiconductor-based spin electronics 共see, e.g., Ref. 1兲 has developed into a prominent branch of present spintronics re- search. In this context spin-orbit 共SO兲 interactions have re- cently received considerable attention since they give rise to rich spin dynamics and a variety of spin phenomena in non- magnetic semiconductors. Though SO couplings have been a subject of continuous research throughout the last decades,^2–8 there is presently a revival in investigating SO effects owing to their role in spin transistors,^9,10 spin interferometers,^11,12 spin filters,^13,14 and spin pumps,^15,16 to name only a few examples. Furthermore, most recently the intrinsic spin Hall effect^17,18 in a SO-coupled system has caused an intense and controversial discussion in the litera- ture. Finally, in spin-based quantum computation SO- induced spin relaxation effects may play a role.¹⁹

The interplay between spin dynamics and confinement ef- fects is particularly intriguing in quantum transport through low-dimensional devices at low temperatures where quantum coherence effects additionally arise. There exist two promi- nent experimental probes for SO effects in quantum trans- port,共i兲characteristic beating patterns in Shubnikov-de Haas oscillations in two-dimensional electron gases with tunable SO coupling, controlled via a back-gate voltage,^20–23and共ii兲 weak antilocalization^24–27 共WA兲, an enhancement of the magnetoconductance at zero magnetic field owing to spin- dependent quantum interference effects. Since systems with- out SO coupling exhibit weak localization共WL兲, i.e., a re- duction in the magnetoconductance, the appearance of WA allows conclusions to be drawn on the SO strength. While WA is fairly well understood for disordered bulk systems,^28–30in recent experiments using ballistic bismuth³¹ and GaAs³² cavities, WA has been employed to study SO- induced spin dynamics and spin relaxation phenomena in confined systems. These measurements are focussed on the interesting interrelation between quantum confined orbital motion and spin evolution and relaxation in clean ballistic quantum dots. In these systems, the elastic mean free path is

exceeding their size, and impurity scattering is replaced by reflections off the system boundaries.

Corresponding efforts in treating SO effects on spectra,^33–36 spin relaxation,^37–39 and the interplay between SO and Zeeman coupling^40,41in quantum dots have also been made on the theoretical side. SO-induced WA in ballistic quantum dots has been studied using random-matrix theory^42,43 共RMT兲 and semiclassical approaches.^44,45While RMT approaches are restricted to quantum dots with corre- sponding chaotic classical dynamics, the semiclassical trans- port theory comprises a much broader class of systems, in- cluding integrable confinement geometries. Related semiclassical techniques have also been applied to spin transmission⁴⁶ and spin relaxation⁴⁷ in quantum dots.

The purpose of the present paper is a detailed exposition and extension of the semiclassical methods of Refs. 44 and 45. The theory to be discussed here unifies two subject areas, the semiclassical description of WL^48–50and the semiclassi- cal treatment of SO interaction.^51–57Compared to the earlier works^44,45,47on the subject, here we give special attention to the differences in spin relaxation along open and closed tra- jectories, analyze the interplay between Rashba and Dressel- haus interaction, and report on the full quantum calculations of spin-dependent transmission and reflection.

This paper is organized as follows: In the introductory Sec. II, using path integrals with spin coherent states we deduce a spin-dependent semiclassical propagator and the corresponding Green function. Our main analytical results are presented in Sec. III. There, on the basis of Green func- tions a semiclassical approximation to the Landauer formula with spin is derived. The semiclassical Landauer formula is then applied to chaotic quantum dots, whereby the quantum corrections to transmission and reflection are calculated. In Sec. IV we discuss how WA is related to the spin evolution.

We define measures for spin relaxation and consider, as an example, the spin relaxation in diffusive systems. In the fol- lowing two sections the general theory is applied to chaotic and integrable quantum dots with Rashba and Dresselhaus SO interaction. In Sec. V a detailed numerical study of the spin relaxation is followed by an analysis of the limit of slow spin dynamics共i.e., extremely weak SO coupling兲. Addition- 1098-0121/2005/72共15兲/155325共18兲/$23.00 155325-1 ©2005 The American Physical Society

ally, we examine a gauge transformation of the spin-orbit Hamiltonian that can be carried out in this limit. The depen- dence of the quantum corrections to transmission and reflec- tion on the SO-coupling strength and magnetic field 共Aharonov-Bohm and Zeeman contributions兲is presented in Sec. VI. There some of the semiclassical numerical results are compared with full-scale numerical quantum calcula- tions.

II. SPIN-ORBIT INTERACTION IN A SEMICLASSICAL THEORY

In this preparatory section we construct a spin-dependent semiclassical propagator⁸¹ and related Green function. It fully describes the system at a given level of approximation and, thus, can serve as a starting point for our derivation of a semiclassical Landauer formula for systems with SO and Zeeman interaction共Sec. III兲.

In the spinless case, the semiclassical propagator is con- ventionally obtained from the path-integral representation of the exact propagator.⁵⁸After the stationary-phase evaluation, which is valid in the semiclassical limit, the classical trajec- tories are selected from all the paths in the integral. In order to include spin into the path integral, a continuous basis of spin states is required. The spin coherent states represent such a basis.^59,60

Following Ref. 60, we define a coherent state of spins

=¹₂,1,…by

兩␨典=共1 +兩␨兩²兲^−sexp共␨^sˆ+兲兩␴^{= −}s典, 共1兲 where ␨ is a complex number that labels the state, sˆ₊=sˆx

+isˆ_y is the spin operator, and 兩␴典 are the eigenstates of sˆ_z with eigenvalues␴^{= −s,}…,s. To each␨corresponds a three- dimensional unit vectorn共␨兲=具␨兩sˆ兩␨典/sthat denotes the spin direction. It is easy to show that␨is a stereographic projec- tion from the unit sphere centered at the origin onto the plane z= 0. The projection is given by 共nx,ny,nz兲哫共Re␨^,

−Im␨, 0兲, where, in particular, the south pole is mapped to

␨= 0. In general, coherent states have the minimal uncer- tainty of sˆ among all spin states and are characterized by three real parameters: the directionn and an overall phase.

共Hence, any state of spin 1 / 2 is coherent.兲 In the current definition, the phase is assigned to each n by Eq. 共1兲, but other phase assignments are possible. Note that the phase of the state兩␨⁼⬁典⬀兩␴⁼s典withn=共0 , 0 , 1兲is not well defined.

However, this manifestation of the fundamental problem of phase assignment⁶¹ does not pose a difficulty in our case, since the final results will be transformed to the 兩␴典 repre- sentation using the projection operators兩␨典具␨兩. The states共1兲 are normalized to unity, but, obviously, not mutually or- thogonal共no more than 2s+ 1 states of spinscan be mutually orthogonal兲. Nevertheless, having the property of resolution of unity,

冕

^{兩␨典具␨兩d␮共␨兲=1ˆ, d␮共␨兲=}␲共1 +^{2s+ 1}兩␨兩²兲²d²␨^, 共2兲 they form an共overcomplete兲basis and enable a path-integral construction.

Let us consider a rather general case of a system with Hamiltonian linear in the spin operatorsˆ,

Hˆ =Hˆ⁰共qˆ,pˆ兲+បsˆ·Cˆ共qˆ,pˆ兲. 共3兲 Hereqˆ andpˆ are thed-dimensional coordinate and momen- tum operators, respectively,Hˆ⁰共qˆ,pˆ兲is the spin-independent Hamiltonian, and បsˆ·Cˆ共qˆ,pˆ兲 describes the SO interaction and the Zeeman interaction with an external共generally inho- mogeneous兲 magnetic field. Utilizing the idea of Refs. 54 and 56, we express the propagator in the combined coordi- nate and spin-coherent-state representation in terms of the path integral,

U共q₂,␨2,q₁,␨1;T兲 ⬅ 具q₂,␨2兩e^{−共i/ប兲Hˆ T}兩q₁,␨1典

=

冕

^D关共2^q␲^兴D关ប兲^dp兴D␮关␨兴exp

再

^{បiW关q,p,␨;T兴}

冎

^.

共4兲 The integration is performed over the paths关q共t兲,p共t兲,␨共t兲兴 in the spin-orbit phase space connecting 共q1,p₁,␨1兲 to 共q2,p₂,␨2兲 in timeT with arbitrary p₁ andp₂. The integra- tion measures are defined by

D关q兴D关p兴

共2␲ប兲^d D␮关␨兴= lim

n→⬁

兿

j=1

n−1dq共t_j兲dp共t_j兲

共2␲ប兲^d d␮„␨共t_j兲…, 共5兲 where tj=jT/n. The Hamilton principal function W=W⁰ +បsW¹ consists of two contributions: the usual classical part,

W⁰关q,p;T兴=

冕

0 T

dt关p·q˙−H⁰共q,p兲兴, 共6兲 and the spin-related part,

W¹关q,p,␨^;T兴=

冕

0 T

dt

冋

^{i共1 +␨␨˙*−兩␨␨*兩2␨˙兲−n共␨兲·C共q,p兲}

册

^{. 共7兲}

Now we can separate the integration over the spin paths in Eq.共4兲, thereby representing the propagator as⁵⁶

U共q2,␨2,q₁,␨1;T兲=

冕

^D关共2^q␲^兴D关ប兲^dp兴K_关q,p兴共␨2,␨1;T兲

⫻exp

再

^{បiW0关q,p;T兴}

冎

^共8兲

with

K_关q,p兴共␨2,␨1;T兲=

冕

^{D␮关␨兴exp}

^兵

^{isW1关q,p,␨;T兴}

^其

^{. 共9兲}

Clearly,K_关q,p兴共␨2,␨1;T兲is a propagator of a system with the time-dependent Hamiltonian Hˆ

关q,p兴共t兲=បsˆ·C_关q,p兴共t兲, where C_关q,p兴共t兲=C共q共t兲,p共t兲兲 is calculated along the path 关q共t兲,p共t兲兴 of the integral共8兲. This Hamiltonian describes a spin, precessing in the time-dependent magnetic field C_关q,p兴共t兲.⁸² Expression 共9兲 for K_关q,p兴共␨2,␨1;T兲 can be inte-

grated explicitly,⁶⁰yielding the usual spin propagator in the basis of coherent states共Appendix A兲.

We proceed by evaluating the path integral 共8兲 in the semiclassical limit W⁰Ⰷប. The integration simplifies con- siderably, if the spin-dependent Hamiltonian is treated as a perturbation, i.e., when

បs兩C共q,p兲兩Ⰶ兩H⁰共q,p兲兩. 共10兲 This condition, assumed for the rest of the paper, is usually fulfilled in experiments based on the semiconductor hetero- structures. According to this requirement, the spin-precession length must be much larger than the Fermi wavelength, how- ever it can be smaller, of order, or greater than the system size. The semiclassical and perturbative regimes can be implemented simultaneously^62,63 by formally letting ប→0 and keeping all other quantities fixed. Then the phase of the integrand in Eq. 共8兲 is a rapidly varying functional, which justifies the use of the stationary-phase approximation. It is crucial thatK_关q,p兴共␨2,␨1;T兲does not depend onប, i.e., it is a slowly varying functional, and, therefore, its effect on the stationary trajectories can be neglected. Thus, the stationary trajectories are the extremals solely of W⁰关q,p;T兴, which means that they are the classical orbits of the spinless Hamil- tonianH⁰. The resulting semiclassical propagator,

U_sc共q2,␨2,q₁,␨1;T兲

=

兺

_␥ ^{K␥共␨2,␨1;T兲C␥exp}

再

^{បiW␥0共q2,q1;T兲}

冎

^{, 共11兲}

is a sum over all classical trajectories ␥⬅关q_␥共t兲,p_␥共t兲兴 of time T from q₁ to q₂. The prefactor C_␥, arising from the stationary-phase integration, is the same as in the spinless case:⁵⁸

C_␥=exp

共

^−i␲4d−i^␲₂␯␥

兲

共2␲ប兲^d/2

冏

^{det␣␤⳵2W}⳵^␥0q共q2␣⳵^2,qq1␤^1;T兲

冏

^1/2,

共12兲 where␯␥is the Maslov index. Although the classical trajec- tories are not affected by the spin motion, the reverse is not true. Indeed, the spin propagator K_␥, computed along the classical trajectories, describes the spin evolution in the ef- fective magnetic field C_␥共t兲=C共q_␥共t兲,p_␥共t兲兲 generated by these trajectories.

The semiclassical Green function is given by the Laplace transform ofU_sc共T兲to the energy domainE,

G共E兲= − i ប

冕

0

⬁

dT e^{i共E+i0+兲T/ប}U_sc共T兲, 共13兲

evaluated in the stationary-phase approximation. As before, K_␥共T兲 does not modify the stationary-phase condition, and the theory without spin can be applied. Moreover, using the resolution of unity 共2兲, the spin propagator can be trans- formed to the usual兩␴典 basis. Finally, we obtain

G_␴⬘␴共q2,q₁;E兲=

兺

_␥ ^{共Kˆ␥兲␴}⬘^␴F_␥exp

再

^{បiS␥0共q2,q1;E兲}

冎

共14兲 with␴^,␴

⬘

^{= −s,…,s. In Eq.共14兲,}␥is a classical trajectory of the HamiltonianH⁰=Ewith the actionS_␥⁰=兰_␥p·dqand time T_␥共E兲=⳵S_␥⁰/⳵^E.Kˆ␥共t兲is the operator form of the spin propa- gator 关Appendix A, Eqs. 共A5兲 and 共A6兲兴, and 共Kˆ

␥兲_␴⬘␴

⬅具␴

⬘

^兩Kˆ␥(T_␥共E兲)兩␴典 is its matrix element. The prefactor is given by

F_␥=C_␥e^{−i共␲/4兲sgn共dT␥/dE兲}, 共15兲 and C_␥ is expressed in terms of the derivatives of S_␥⁰共q2,q₁;E兲 共Ref. 58兲.

III. SEMICLASSICAL LANDAUER FORMULA WITH SPIN The semiclassical Landauer formula with spin, derived below, is the main analytical result of this paper. It forms the basis for the subsequent semiclassical treatment of the spin- dependent transport in two-dimensional systems.

A. Derivation of the formula

We start from the standard共quantum兲Landauer formula, that relates the conductance 共e²/h兲T of a sample with two ideal leads to its transmission coefficientT共Ref. 64兲. Assum- ing that the leads support N and N

⬘

open channels 共not counting the spin degeneracy兲, respectively, the transmission can be expressed as the sum

T=

兺

n=1 N⬘

m=1

兺

N

兺

␴⬘^,␴=−s

s

兩t_n␴⬘,m␴兩². 共16兲

Here tn␴⬘^,m␴ is the transmission amplitude between the in- coming channel 兩m␴典 共with spin projection␴兲and the out- going channel 兩n␴

⬘

^典 belonging to different leads. We shall also consider the reflection coefficient

R=

兺

n,m=1

N

兺

␴⬘^,␴=−s

s

兩rn␴⬘^,m␴兩², 共17兲

where the reflection amplitudern␴⬘^,m␴is defined for the chan- nels of the same lead. The transmission and reflection satisfy the normalization condition

T+R=共2s+ 1兲N 共18兲 that follows from the unitarity of the scattering matrix.

Consider, as a model for a 共large兲 quantum dot, a two- dimensional cavity 共billiard兲 with hard-wall leads. The par- ticle in the cavity is subjected to the SO and Zeeman inter- action of the form 共3兲. Semiclassical expressions for the transition amplitudes in the spinless case were derived in Refs. 48 and 49 by projecting a semiclassical Green function onto the lead eigenstates, while integrating over the lead cross sections in the stationary-phase approximation. For a particle with spin we implement this procedure using the

semiclassical Green function共14兲. In the semiclassical limit of large action,S_␥⁰Ⰷប, the spin-propagator element共Kˆ

␥兲_␴⬘␴ does not shift the stationary point. In the resulting expres- sion,

tn␴⬘^,m␴=

兺

␥共¯n,m¯兲

共Kˆ

␥兲_␴⬘␴A_␥exp

冉

បⁱS_␥

冊

^{, 共19兲}

the only effect of spin is to weight the contribution of each trajectory in the sum with the respective matrix element of Kˆ_␥. In Eq.共19兲␥共n¯,m¯兲is any classical trajectory of energyE that enters共exits兲the cavity at a certain angle⍜m¯ 共⍜¯n兲mea- sured from the normal at a lead’s cross section.⁸³The angles are determined by the transverse momentum in the leads:

sin⍜m¯=m¯␲^{/kw and sin}⍜n¯=¯n␲^/kw

⬘

^{, where k} is the wave number andwandw

⬘

are the widths of the entrance and exit leads. The action for a trajectory of lengthL_␥is S_␥=បkL_␥. The prefactor is given by

A_␥= −

冑

2ww^␲ប

⬘

sgn共n¯兲sgn共m¯兲 兩cos⍜¯ncos⍜m¯M₂₁^␥兩^1/2

⫻exp

冋

^{ik共sin⍜m¯y− sin⍜n¯y}

^⬘

^兲−i␲2

^冉

^␮␥−12

^冊 册

^,

共20兲 whereM₂₁^␥ is an element of the stability matrix共as defined, e.g., in Ref. 65兲,y共y

⬘

兲 is the coordinate on the lead’s cross section at which the orbit␥ ^enters共exits兲the cavity, and␮_␥ is the modified Morse index.⁴⁹Substituting the sum共19兲and the corresponding result forrn␴⬘^,m␴ in Eqs.共16兲and共17兲we obtain the semiclassical approximation for the total transmis- sion and reflection,

共T,R兲=

兺

_nm

兺

␥共¯n,m¯兲

兺

␥⬘^{共n¯,m¯兲}

M_␥,␥⬘A_␥A_␥^*_⬘exp

冉

បⁱ共S_␥−S_␥⬘兲

冊

^.

共21兲 Here in the case of transmission共reflection兲the paths␥^and

␥

⬘

connect different leads 共return to the same lead兲. In this expression each orbital contribution is weighted with the spin modulation factor

M_␥,␥⬘⬅Tr共Kˆ

␥Kˆ

␥⬘

†兲, 共22兲

where the trace is taken in spin space. Equations 共21兲 and 共22兲generalize the semiclassical Landauer formula^48,49to the case of spin-dependent transport.

B. Leading semiclassical contributions for a spinless particle In the semiclassical limit the phases in Eq.共21兲are rap- idly varying functions of energy, unless␥and␥

⬘

^{have equal} or nearly equal actions. Therefore, if one calculates the trans- mission and reflection averaged over a small energy window, most of the terms in the double sum will vanish. In the fol- lowing, we review the leading contributions for a spinless system共M_␥,␥⬘⬅1兲with time-reversal symmetry:

共i兲 The classical part consists of the terms with ␥

⬘

⁼␥

共Ref. 66兲. Their fast-varying phases cancel 共including the phase in the prefactor兲. For a classically ergodic共in particu- lar, chaotic兲system one finds⁵⁰

T_cl^共0兲= NN

⬘

N+N

⬘

^{, Rcl共0兲=} N²

N+N

⬘

^共23兲

共the superscript refers to zero spin and zero magnetic field兲. This result can be obtained using the sum rule⁵⁰

␥共

兺

¯n,m¯兲兩A_␥兩²␦共L−L_␥兲 ⯝ 共N+N

⬘

兲⁻¹P_L共L兲. 共24兲 It implies that the length L of the classical trajectories is distributed according to

PL共L兲 ⯝ 1

L_escexp

冉

⁻L^L_esc

冊

^{, 共25兲}

in other words, the probability for a particle to stay in an open chaotic cavity decreases exponentially with time. The average escape length is given by

L_esc= ␲^Ac

w+w

⬘

⁼ kAc

N+N

⬘

^{, 共26兲}

where A_c is the area of the cavity. It is assumed that L_esc ⰇLb, where

L_b=␲^Ac/P_c 共27兲 is the average distance between two consecutive bounces at the boundaries⁶⁷andP_cis the perimeter. In an arbitrary bil- liard the last expression is true if the average is taken over the ensemble of chords with random initial position and boundary component of the velocity. In ergodic billiards the average can, alternatively, be calculated along almost any trajectory.

共ii兲 The diagonal quantum correction is defined for re- flection only. It contains the terms with n=m and␥

⬘

^=␥−1, where␥⁻¹is the time reversal of␥共Ref. 49兲.⁸⁴Clearly, when n and m are different channels, the orbits ␥共¯n,m¯兲 and

␥

⬘

共¯n,m¯兲cannot be the mutual time reversals, since reversing the time would exchange¯nandm¯. Again, the two actions are equal, and the result for an ergodic system without spin reads⁵⁰

␦^Rdiag共0兲 = N

N+N

⬘

^{. 共28兲}

共iii兲 The loop contribution consists of pairs of long orbits that stay close to each other in the configuration space. One orbit of the pair has a self-crossing with a small crossing angle ␧, thus forming a loop, its counterpart has an anticrossing.^77,85Away from the crossing region the orbits are located exponentially close to each other, they are related by time reversal along the loop and coincide along the tails^50,68共Fig. 1兲. The action difference for these orbits is of second order in␧. For spinless chaotic systems with hyper- bolic dynamics the loop terms in Eq.共21兲yield⁵⁰

␦Tloop共0兲 = − NN

⬘

共N+N

⬘

兲², ␦Rloop共0兲 = − N共N+ 1兲

共N+N

⬘

兲². 共29兲

From here on, we will work in the limitN,N

⬘

^{Ⰷ1. In this} semiclassical regime共in the leads兲the classical contribution 共23兲 共of the orderN兲is much greater than the quantum cor- rections 共28兲 and共29兲 共of the orderN⁰兲, while higher-order loop corrections共of orderN⁻¹and smaller兲can be neglected.

We note that the normalization is preserved order by order, Tcl共0兲+Rcl共0兲=N, 共30兲

␦Rdiag共0兲 +␦Rloop共0兲 +␦Tloop共0兲 =O共N⁻¹兲. 共31兲

C. Spin-dependent quantum corrections to transmission and reflection

We now compute the spin modulation factor for the lead- ing contributions to the energy-smoothedTandR, identified in Sec. III B. First, the case with time-reversal symmetry⁸⁶is considered.

共i兲For the classical part we find, using the unitarity ofKˆ

␥, that the modulation factor M_␥,␥= Tr共Kˆ

␥Kˆ

␥†兲= 2s+ 1 reduces to the trivial spin degeneracy.

共ii兲For the diagonal quantum correction the result is M_␥,␥⁻¹= Tr共Kˆ

␥2兲. 共32兲

It was taken into account thatKˆ

␥⁻¹

† =Kˆ

␥, which follows from the relationC_␥⁻¹共t兲= −C_␥共T_␥−t兲and Eq.共A5兲.

共iii兲 Assuming that the trajectories forming a loop pair 共Fig. 1兲coincide along the tailst1,t2 and are mutually time- reversed along the loop l, thereby neglecting the crossing region, we can represent the propagators asKˆ

␥=Kˆ

t2Kˆ

lKˆ

t1and Kˆ

␥⬘=Kˆ

t2Kˆ

l⁻¹Kˆ

t1. Hence, the modulation factor M_␥,␥⬘= Tr共Kˆ

l

2兲. 共33兲

is independent of the tails.

In the presence of a magnetic field the time-reversal sym- metry is broken, and the preceding results should be ad- justed. In this paper we consider a constant, uniform, arbi- trarily directed magnetic fieldB. Its componentBznormal to the cavity is assumed to be weak enough,⁸⁷ so as not to change the classical trajectories in Eq.共21兲, but only modify the action difference by the Aharonov-Bohm共AB兲phase. We define the AB modulation factor

␸_␥,␥⬘⬅exp

冉

បⁱ⌬共S_␥−S_␥⬘兲

冊

^{= exp}

冉

^i4␲⌽^AB0 ^z

冊

^{. 共34兲}

Here, for a pair of trajectories ␥ ^and ␥

⬘

from the diagonal 共loop兲 contribution, A⬅兰_␥共l兲A·dl/Bz is the effective en- closed area, where the integral of the vector potential A is taken along ␥ 共its loop part l兲, and ⌽0=hc/e is the flux quantum.

With the Zeeman interaction included, Eqs.共32兲and共33兲 for the spin modulation factor are no longer valid. In fact, if we distinguish the SO and Zeeman terms in the effective magnetic fieldC_␥共t兲=C_␥^SO共t兲+C_␥^Z共t兲, the diagonal共and, cor- respondingly, the loop兲 modulation factor can be written in the form

M_␥,␥⁻¹共B兲= Tr共Kˆ

␥Kˆ

␥

˜兲, 共35兲

where^˜␥ is a fictitious trajectory producing the field C˜_␥共t兲

=C_␥^SO共t兲−C_␥^Z共t兲. Clearly, in the absence of SO coupling, we haveM_␥,␥⁻¹共B兲= 2s+ 1, i.e., the Zeeman field alone does not affect the modulation factor.

In Refs. 49 and 50 the quantum corrections to transmis- sion and reflection in the presence of magnetic field were calculated for an ergodic system. We extend their approach to a system with SO interaction. To this end, we consider the generalized modulation factor

共M␸兲_␥,␥⬘⬅M_␥,␥⬘␸␥,␥⬘. 共36兲 The diagonal contribution can be computed using the sum rule共24兲. First, one averages共M␸兲_␥,␥⬘for the time-reversed pairs of trajectories and loops of a given lengthL. Thus, the average is performed over the ensemble of almost closed orbits. This restriction proves very important, since the aver- age modulation factor for closed and open trajectories is dif- ferent 共see Sec. V兲. The average modulation factor M␸共L;B兲is then further weighted with the length distribu- tion共25兲. It can be shown that for the loop contribution in a hyperbolic system with a single Lyapunov exponent holds effectively the same procedure 共see Ref. 50 and Appendix B兲. Hence, the diagonal and the loop relative quantum cor- rections to reflection and transmission are equal and given by

␦Rdiag共B兲

␦R_diag^共0兲 =␦Rloop共B兲

␦R_loop^共0兲 =␦Tloop共B兲

␦T_loop^共0兲 =具M␸共B兲典L

⬅ 1

L_esc

冕

0

⬁

dL e^−L/LescM␸共L;B兲. 共37兲 The normalization condition 共18兲 is preserved due to Eq.

共31兲.

When the SO and Zeeman interactions are absent, the average modulation factor 共2s+ 1兲␸^¯共L;B兲 in a chaotic sys- tem can be analytically estimated using the Gaussian distri- bution of enclosed areas⁴⁹

P_A共A;L兲 ⯝ 1

冑

2␲^A0 2L/Lb

exp

冉

^−2AA02L/L² b

冊

^{. 共38兲}

It depends on a system-specific parameterA₀, a typical area enclosed by an orbit during one circulation. This distribution FIG. 1.共Color online兲Pair of orbits with a loop.共Neglecting the

crossing region, we distinguish between the tailst1,t2, and the loop l.兲

does not depend on the incoming and outgoing channel num- bers and is valid for both closed and open trajectories. The average共37兲yields^49,50

具M␸共B兲典L= 2s+ 1 1 +B˜²L_esc/L_b

, 共39兲

whereB˜= 2

冑

2␲^BzA₀/⌽0. Thus the quantum corrections have a Lorentzian dependence on the magnetic field. The result 共39兲 is specific to chaotic systems as it depends on L_esc—such a parameter is not relevant to extended disordered systems, while for regular billiardsPL共L兲 共introduced earlier兲 is usually a power law.⁴⁹The increase of reflection共decrease of transmission兲for B_z= 0 constitutes the effect of weak lo- calization. A magnetic field destroys the time-reversal sym- metry and, thereby, the interference between the mutually time-reversed and loop pairs of paths, thus diminishing the quantum corrections.

The SO interaction may turn the constructive interference between the orbit pairs into destructive one. Since the sign of the quantum corrections in this case would be reversed, one speaks of weak antilocalization. In the following sections we study the transition from WL to WA and the related question of spin relaxation.

IV. SPIN RELAXATION A. General discussion

Equation 共37兲 demonstrates that the modulation factor M␸共L;B兲is a key to calculating the quantum corrections to the conductance. Hence, we will first examineM␸^{in detail.}

As a function of length, this quantity contains information about the average spin evolution along the trajectories of the system. Here, by the spin evolution along a trajectory␥^we mean the change of the spin propagatorKˆ

␥共t兲. According to Appendix A, it can be written in the form关Eq.共A6兲兴

Kˆ

␥共t兲= exp关−isˆ·␩␥共t兲兴, 共40兲 and, thus, depends on three real parameters, the rotation angle ␩␥共t兲 and the rotation axis given by the unit vector m_␥共t兲⬅␩␥共t兲/␩␥共t兲. Alternatively, one can parametrize Kˆ using the elements of the corresponding SU共2兲 matrix关Eq.␥

共A3兲兴,

W_␥共t兲=e^{−i␴·␩␥共t兲/2}=

冉

^{−ab␥共t兲}␥*共t兲 ^ba^␥_␥^*共t兲共t兲

冊

^{, 共41兲}

which are restricted by the condition detW_␥=兩a_␥兩²+兩b_␥兩²= 1.

The two parametrizations are related by Eq. 共A4兲. Clearly, W_␥is the matrix representation ofKˆ

␥for spins= 1 / 2. Instead of W_␥共t兲, we can consider the evolution of the spinor ␺␥共t兲

⬅(a_␥共t兲, −b_␥^*共t兲)^T, starting from the spin-up state ␺␥共0兲

=共1 , 0兲^T. It is characterized by the spin direction n_␥共t兲

=关␺_␥共t兲兴^T␴␺_␥共t兲 and the overall phase 共␴ is the vector of Pauli matrices兲. For spins⬎1 / 2 these are the direction and the phase of a coherent state共see Sec. II兲. Note that n_␥共t兲 results from the rotation ofn_␥共0兲=共0 , 0 , 1兲by the angle␩␥共t兲

about m_␥共t兲. Sometimes it is convenient to representW_␥共t兲 by a trajectory on the three-dimensional unit sphereS³. For this purpose we define a four-dimensional unit vector

␰_␥共t兲=„− Imb_␥共t兲,− Reb_␥共t兲,− Ima_␥共t兲,Rea_␥共t兲…

=

冉

^{m␥共t兲sin␩␥2共t兲,cos␩␥2共t兲}

冊

^{. 共42兲}

The trajectory starts at the “north pole,” i.e., ␰_␥共0兲

=共0 , 0 , 0 , 1兲.

Using the propagator matrix 共41兲, the modulation factor Eqs.共32兲and共33兲for spin-1 / 2 can be expressed as

M= Tr共W²兲= 4␰42− 2 = 2 cos␩^, 共43兲 where ␰4 is the fourth component of ␰ 共in Appendix C an arbitrary s is considered兲. To simplify the notation, we dropped the subscripts labeling the trajectory, and the timet is the trajectory or loop time.

For long orbits one expects that the spin state becomes completely randomized due to SO interaction, if the particle motion is irregular. This means that all points ␰苸S³ are equally probable, and, on average, ␰4

2= 1 / 4 in the limit L

→⬁. Hence, for B= 0 the modulation factor M¯ 共L兲

⬅M␸共L; 0兲 changes with L from the positive valueM¯ 共0兲

= 2 to the negative asymptotic valueM¯ 共⬁兲= −1共cf. Ref. 69兲.

The stronger the SO interaction, the shorter the length scale L_M of this change. If L_escⰆL_M, i.e., the particle quickly leaves the cavity, or the SO interaction is weak, then the relative quantum corrections共37兲are positive, giving rise to WL. In the opposite case of strong SO interaction or long dwell times, the relative quantum corrections are negative, leading to WA. For an arbitrary spin, M¯ 共L兲 changes from M¯ 共0兲= 2s+ 1 to M¯ 共⬁兲=共−1兲^2s 共Appendix C兲. Thus, WA cannot be observed for an integer spin, at least, for L_esc ⰇL_M关M¯ 共L兲can, in principle, become negative at interme- diate lengths兴.

For the rest of the paper we will consider the physically most important case of spin s= 1 / 2. If the Zeeman interac- tion is included, then Eq.共35兲yields

M共B兲= 4␰4^˜␰4− 2␰^·˜␰^, 共44兲 where^˜␰belongs to the fictitious trajectory␥^˜. In the absence of Zeeman coupling, the vectors␰^and˜␰ coincide. Then the negative second term in Eq.共44兲is responsible for the WA, if the first term is, on average, small enough due to SO inter- action. An admixture of a moderate Zeeman coupling de- stroys the correlation between␰^and˜␰, thereby reducing the average product ␰^·˜␰. Thus, an external magnetic field sup- presses WA in two ways: the AB flux breaks down the con- structive interference between the orbital phases and the Zee- man interaction affects the spin modulation factor. As we know, the former mechanism inhibits the WL, as well.

The spin propagator Kˆ

␥共t兲 can be used not only in the calculation of the quantum corrections共37兲—it also provides information about the spin relaxation along classical trajec-

tories, which is of separate interest. The relaxation of the spin direction can be described by the z component of the vectorn_␥,

nz= 2共␰32+␰42兲− 1. 共45兲 The ensemble average nz共L兲 varies from nz共0兲= 1 to nz共⬁兲

= 0, if the memory of the initial spin direction is completely lost for long orbits. The typical length scale of this decay can be different fromL_M, because M¯ 共L兲 depends on the phase of the spin state, as well as on its direction. Moreover, the length scale ofn_z共L兲, as defined by Eq.共45兲, depends on the choice of the quantization axis. An invariant measure of the spin relaxation is given by␰4

2共L兲or, equivalently, by M¯ 共L兲.

The different relaxation rates of nz共L兲 and M¯ 共L兲 are ob- served in two-dimensional systems with the Rashba and the Dresselhaus SO coupling共see Secs. IV B and V兲.

B. Example: Diffusive systems

In three-dimensional extended diffusive conductors⁸⁸ the directions of the effective magnetic field C_␥共t兲 before and after a scattering event can be assumed uncorrelated.³⁰ We model this by keeping兩C兩= const and changing the direction ofCrandomly at identical time intervals equal to the elastic scattering time␶. The spin propagator for the jth time inter- val isKˆ

j= exp共−isˆ·mj兩C兩␶兲,j= 1,2,…, wheremjis a random unit vector. The position onS³after the first time interval is, according to Eq. 共42兲, ␰共␶兲=共m1sin兩C兩␶/ 2 , cos兩C兩␶/ 2兲.

Thus, a trajectory on the sphere, starting at the “north pole,”

traverses an arc of length 兩C兩␶/ 2 along a randomly chosen great circle. During the second time interval, the trajectory starts at ␰共␶兲 and moves along another random great-circle segment, and so on. Clearly,␰共t兲follows a random walk on S³. In the continuous limit兩C兩␶Ⰶ2␲, its probability density satisfies a diffusion equation. Solving this equation共Appen- dix D兲we find that the average modulation factor for trajec- tories of timet,

Mdiff,3D共t兲= 3e^{−共1/3兲兩C兩2␶t}− 1, 共46兲 and the average spin polarization,

共nz兲diff,3D共t兲=e^{−共1/3兲兩C兩2␶t}, 共47兲 exhibit the same relaxation rate. Note that Eq. 共37兲 is not valid in diffusive systems. The modulation factor 共46兲 is equivalent to the result of Eq.共10.12兲of Ref. 30.

In two-dimensional diffusive systems with Rashba or Dresselhaus interaction it is reasonable to assume thatCac- quires a random direction in a two-dimensional plane. In this case the walk onS³ is not fully random. As our numerical simulations show共Fig. 2兲, the modulation factor is reason- ably well described by Eq.共46兲. However, the off-plane po- larizationn_z共t兲relaxes faster in two dimensions关cf. Eqs.共34兲 and共35兲of Ref. 70兴. This is not surprising, since, obviously,

␰3共␶兲⬅0 in 2D, but not in 3D.

V. RASHBA AND DRESSELHAUS INTERACTION:

SPIN RELAXATION A. Effective magnetic field

We apply the general theory of the preceding sections to ballistic quantum dots with Rashba³ and Dresselhaus⁴ SO interaction. Both contributions are usually present in GaAs/ AlGaAs heterostructures. Their strength ratio can be experimentally varied, e.g., by tuning the Rashba SO strength through an additional gate voltage.²¹When the two- dimensional electron gas lies in the 共001兲 plane of a zinc- blende lattice, the effective magnetic fieldCˆ=Cˆ

R+Cˆ

Din the Hamiltonian共3兲consists of

Cˆ

R=2␲

⌳R

vˆ⫻e_z, 共48兲

Cˆ

D= 2␲

⌳D

共vˆ_xex−vˆ_yey兲, 共49兲 where thexandyaxes are chosen along the关100兴and关010兴 crystallographic directions, respectively, and vˆ=共pˆ−eA/c兲/

Mis the共Fermi-兲velocity operator depending on the effective mass M. In Eq. 共48兲 关Eq. 共49兲兴 the Rashba 共Dresselhaus兲 interaction, usually characterized by the constant␣R共␣D兲, is measured in terms of the inverse spin-precession length

⌳R−1共D兲

=␣R共D兲M/␲ប². In billiards the natural dimensionless parameter is

␪R共D兲= 2␲^Lb/⌳R共D兲. 共50兲 It signifies the mean spin-precession angle per bounce if only one type of SO interaction is present.

As can be seen from Eq.共48兲, the effective Rashba mag- netic field CR共t兲 generated by a particular trajectory points perpendicular to the velocity v共t兲. The directions of the Dresselhaus fieldCD共t兲 andv共t兲are symmetric with respect to thexaxis关Eq.共49兲兴. Hence,CR共t兲andCD共t兲always point FIG. 2.共Color online兲Average spin modulation factorM¯共t兲and spin projectionn_z共t兲 in diffusive systems. The effective magnetic fieldC共t兲changes its direction randomly at equal time intervals␶. Its magnitude is kept constant and is equal to 0.3/␶in this example.

The analytical expressions 共46兲 and 共47兲 共solid curves兲 are com- pared with the results of numerical simulations for C in two 共dashed-dotted curves兲and three共circles兲dimensions. The average was performed over 10⁵random sequences共“trajectories”兲.

symmetrically with respect to the 关11¯0兴 direction, labeled here byX共Fig. 3兲. As a consequence, the total fieldC共t兲 is reflected aboutX under the exchange ⌳R↔⌳Din Eqs. 共48兲 and共49兲. This means that the modulation factor M共t兲 and the polarization projectionsn_z共t兲andn_X共t兲are preserved un- der this transformation. For example, systems with only Rashba or only Dresselhaus interaction have identical spin evolution, if the coupling strengths are the same.

It is sometimes convenient to work in the coordinate frame ofXandY=关110兴 共cf. Ref. 42兲. The projections of the effective magnetic field on these axes are given by

Cˆ_X= 2␲^vˆY/⌳X, Cˆ_Y= 2␲^vˆX/⌳Y, 共51兲 where

⌳X

−1=⌳D

−1+⌳R

−1, ⌳Y

−1=⌳D

−1−⌳R

−1 共52兲

are the effective inverse precession lengths. As above, we can define dimensionless parameters␪X共Y兲= 2␲^Lb/⌳X共Y兲.

B. Numerical study

The computation of the spin evolution in billiard cavities is relatively straightforward, since the classical trajectories there are sequences of straight segments. If only the uniform Rashba and Dresselhaus interaction is present, the effective magnetic fieldC_jis constant along thejth segment共the seg- ment velocityvjis constant, moreover, its magnitudevis the same for all j due to the energy conservation兲. The spin- propagator matrix共41兲for a trajectory,

W⬅e^{−i␴·␩/2}=W_l¯W₁, 共53兲 is a product of the respective matrices

W_j⬅e^{−i␴·␩j/2}=e^{−i␴·Cjtj/2} 共j= 1,…,l兲 共54兲 for thel orbit segments. In practice, it is convenient to re- move the velocity dependence in Eqs.共51兲by using the dis- placement⌬rj⬅⌬XjeX+⌬YjeY=vjtj instead of the segment timetj. Thereby the rotation vector can be expressed as

␩j= 2␲

冉

^{⌬Y⌳XjeX+⌬X⌳YjeY}

冊

^{. 共55兲}

It follows from this equation that rescaling of the system size and the spin-precession lengths by the same factor does not

change the spin relaxation. In other words, given the shape of the billiard, the averagesnzandM¯ as functions ofL/Lb

共computed below兲depend only on the angles ␪R and␪D. We performed a systematic numerical study of spin relax- ation for several billiard geometries 共Fig. 4兲 representative for systems with chaotic and integrable classical dynamics.

The desymmetrized Sinai 共DS兲 billiard, the desymmetrized diamond⁷¹ 共DD兲 billiard, and the desymmetrized Bunimov- ich 共DB兲 stadium billiard represent chaotic cavities. The quarter circle共QC兲, rectangle, and circle are integrable. The average spin relaxation is computed for the closed versions of these billiards.

Figures 5 and 6 depict the average spin relaxation de- scribed byn_z共L兲and its invariant counterpart,M¯ 共L兲, for the chaotic DS and integrable QC billiard. The average is per- FIG. 3.共Color online兲The Rashba effective fieldC_Ris normal

to the velocityv, while the directions of the Dresselhaus fieldC_D andvare symmetric with respect to关100兴. Thus, the directions of

C_RandC_Dare symmetric with respect to关11¯0兴. FIG. 4. Billiard geometries: desymmetrized Sinai共DS兲billiard, desymmetrized diamond共DD兲billiard, desymmetrized Bunimovich 共DB兲stadium billiard, quarter circle共QC兲, rectangle, and circle. The leads are numbered for future reference.

FIG. 5.共Color online兲Average spin projectionn_z共L兲and modu- lation factor M¯共L兲 for the closed chaotic DS共solid curves兲 and integrable QC共dashed curves兲billiard共see Fig. 4兲. Each data point represents the average over 50 000 open trajectories with random initial values at the boundary. The numerical results for a two- dimensional extended diffusive system 共dashed-dotted curves兲, whereL_bis identified with the elastic mean free pathv␶, are shown for comparison. The SO-coupling strength is␪R/ 2␲= 0.2 and ␪D

= 0.

formed over ensembles of open共Fig. 5兲and closed共Fig. 6兲 trajectories of length L starting at random position at the boundary with a random boundary component of the veloc- ity. The strength of the Rashba interaction is chosen as

␪R/ 2␲= 0.2, and the Dresselhaus interaction is absent 共or, equivalently,␪D/ 2␲^{= 0.2 and} ␪R= 0兲.

In Fig. 5, the numerical results for an extended two- dimensional diffusive system, whereLbis identified with the elastic mean free path v␶, are shown for comparison. We observe that on the scale ofL⬃Lb the spin relaxation is the same in all three examples. Indeed, before the first collision with the boundary or a scatterer, the particle moves along a straight line, irrespective of the system it belongs to. On longer length scales relaxation in an extended diffusive sys- tem is much stronger than in confined systems. Moreover, in the integrable billiard saturation takes place.⁸⁹ We also note that n_z共L兲 in the chaotic billiard, similarly to the diffusive system 共Sec. IV B兲, relaxes to its asymptotic value faster thanM¯ 共L兲.

The ensemble of closed orbits共Fig. 6兲is responsible for the quantum corrections to transmission and reflection共Sec.

III兲. We find that the relaxation in this case is much slower than for the ensemble of arbitrary trajectories. Remarkably, the spin projection nz and the rescaled modulation factor 共M¯ + 1兲/ 3 are hardly distinguishable.

The spin evolution in several chaotic systems is compared in Figs. 7 and 8 for the ensembles of open and closed trajec- tories, respectively. All the billiards show a qualitatively similar behavior. The DS and DD billiards have about the same relaxation rate. In the DB billiards the relaxation rate grows continuously, starting from zero, as the ratio of the upper straight side to the radius increases.

In Fig. 9 the modulation factor averaged over the open trajectories is presented for the integrable QC and rectangle billiard. Both systems are characterized by the saturation of spin relaxation and persistent long-time oscillations. The saturation level decreases down to −1 as the SO coupling becomes stronger. The circle billiard, which shows a non-

typical relaxation pattern, is considered in Sec. V C.

When the Rashba and the Dresselhaus couplings work simultaneously,^78–80they mutually counteract their effects on the spin relaxation⁹⁰ 共Fig. 10兲. In the extreme case⌳R=⌳D, i.e.,⌳Y−1= 0, the effective fieldCis always parallel to theX axis⁹¹ 共Fig. 3兲. Hence, the propagator matrices in Eq. 共53兲 commute, and the rotation vector becomes

␩⁼

兺

j=1 l

␩j= 2␲共⌬Y/⌳X兲eX, 共56兲

where⌬Y=兺^l_j=1⌬Yjis theY displacement for the trajectory.

According to Eq.共43兲, the modulation factor is then M= 2 cos共2␲兩⌬Y兩/⌳X兲. 共57兲 On the long-length scale LⰇL_b, 兩⌬Y兩 varies from orbit to orbit between 0 and the system size. Clearly, the averageM¯ should be independent of the orbit lengthL. This explains the saturation in Fig. 10. The saturation level decreases from 2 to 0 as ␪R changes from 0 to ⬁. For closed orbits we have FIG. 6. 共Color online兲 Average spin projection n_z共L兲 共dotted

curves兲and rescaled modulation factor关M¯ 共L兲+ 1兴/ 3共solid curves兲 for the closed chaotic DS and integrable QC billiard. Each data point represents the average over 5000 closed trajectories started at random at the boundary. The SO-coupling strength is␪R/ 2␲= 0.2 and␪D= 0.

FIG. 7.共Color online兲The average modulation factorM¯共L兲for the closed chaotic DS, DD, and DB billiards. In the latter case the ratio of the upper straight side to the radius was taken to be 1共DB, 1兲 and 3 共DB, 3兲. Each data point represents the average over 50 000 open trajectories started at random at the boundary. The SO-coupling strength is ␪R/ 2␲= 0.1 共four upper curves兲 and 0.2 共four lower curves兲, while␪D= 0.

FIG. 8.共Color online兲Same as in Fig. 7, but for the ensemble of closed orbits started at random at the boundary. Each data point represents the average over 5000 trajectories.